Question

Two springs have spring constants
$k_{1} = (40 \pm 0.4)$ N/m and
$k_{2} = (60 \pm 0.6)$ N/m.
If they are connected in parallel, the percentage error in the equivalent spring constant is:

• A. 0.5%
• B. 1%
• C. 0.75%
• D. 1.5%

Hint:

Remember the rules for error propagation: For addition or subtraction ($Z = A \pm B$), absolute errors add ($\Delta Z = \Delta A + \Delta B$). For multiplication or division ($Z = A \cdot B$ or $Z = A/B$), relative (or percentage) errors add ($\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$). For powers ($Z = A^n$), the relative error is multiplied by the power ($\frac{\Delta Z}{Z} = n \frac{\Delta A}{A}$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given two spring constants with their absolute errors. The springs are connected in parallel. We need to find the percentage error in their equivalent spring constant.
Step 2: Key Formula or Approach:
1. When springs are connected in parallel, the equivalent spring constant is the sum of the individual constants:
$k_{eq} = k_{1} + k_{2}$
2. When two quantities are added, their absolute errors add up:
$\Delta k_{eq} = \Delta k_{1} + \Delta k_{2}$
3. The percentage error in a quantity $X$ is given by:
Percentage Error = $\left( \frac{\Delta X}{X} \right) \times 100%$
Step 3: Detailed Explanation:
Given values are:
$k_{1} = 40$ N/m with $\Delta k_{1} = 0.4$ N/m.
$k_{2} = 60$ N/m with $\Delta k_{2} = 0.6$ N/m.
First, calculate the equivalent spring constant, $k_{eq}$:
\[ k_{eq} = k_{1} + k_{2} = 40 + 60 = 100 \text{ N/m} \] Next, calculate the absolute error in the equivalent spring constant, $\Delta k_{eq}$:
\[ \Delta k_{eq} = \Delta k_{1} + \Delta k_{2} = 0.4 + 0.6 = 1.0 \text{ N/m} \] Finally, calculate the percentage error in the equivalent spring constant:
\[ \text{Percentage Error} = \left( \frac{\Delta k_{eq}}{k_{eq}} \right) \times 100% \] \[ \text{Percentage Error} = \left( \frac{1.0}{100} \right) \times 100% = 1% \] Step 4: Final Answer:
The percentage error in the equivalent spring constant is 1%.